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Properties

Label 18.0.36448868215...4963.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 37^{16}$
Root discriminant $386.15$
Ramified primes $3, 37$
Class number $308160531$ (GRH)
Class group $[333, 925407]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8020417344913, 0, 0, 53409637566, 0, 0, -251077338, 0, 0, 37068931, 0, 0, 390942, 0, 0, -1332, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 1332*x^15 + 390942*x^12 + 37068931*x^9 - 251077338*x^6 + 53409637566*x^3 + 8020417344913)
 
gp: K = bnfinit(x^18 - 1332*x^15 + 390942*x^12 + 37068931*x^9 - 251077338*x^6 + 53409637566*x^3 + 8020417344913, 1)
 

Normalized defining polynomial

\( x^{18} - 1332 x^{15} + 390942 x^{12} + 37068931 x^{9} - 251077338 x^{6} + 53409637566 x^{3} + 8020417344913 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-36448868215394073947142909926060511831115554963=-\,3^{45}\cdot 37^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $386.15$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(999=3^{3}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{999}(1,·)$, $\chi_{999}(515,·)$, $\chi_{999}(715,·)$, $\chi_{999}(593,·)$, $\chi_{999}(275,·)$, $\chi_{999}(340,·)$, $\chi_{999}(343,·)$, $\chi_{999}(602,·)$, $\chi_{999}(860,·)$, $\chi_{999}(736,·)$, $\chi_{999}(419,·)$, $\chi_{999}(490,·)$, $\chi_{999}(238,·)$, $\chi_{999}(692,·)$, $\chi_{999}(821,·)$, $\chi_{999}(884,·)$, $\chi_{999}(700,·)$, $\chi_{999}(766,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{1}{7}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{259} a^{9} + \frac{3}{7} a^{3}$, $\frac{1}{259} a^{10} + \frac{3}{7} a^{4}$, $\frac{1}{259} a^{11} + \frac{3}{7} a^{5}$, $\frac{1}{1492099} a^{12} - \frac{9}{5761} a^{9} - \frac{22}{40327} a^{6} - \frac{656}{5761} a^{3} + \frac{557}{40327}$, $\frac{1}{1492099} a^{13} - \frac{9}{5761} a^{10} - \frac{22}{40327} a^{7} - \frac{656}{5761} a^{4} + \frac{557}{40327} a$, $\frac{1}{55207663} a^{14} - \frac{9}{213157} a^{11} - \frac{80676}{1492099} a^{8} - \frac{35222}{213157} a^{5} + \frac{14184}{40327} a^{2}$, $\frac{1}{4442339626011390419729} a^{15} - \frac{8261944566453}{120063233135442984317} a^{12} - \frac{220070812485212564}{120063233135442984317} a^{9} + \frac{771231255264965895}{120063233135442984317} a^{6} + \frac{1030161062570439081}{3244952246903864441} a^{3} - \frac{1358388263151521128}{3244952246903864441}$, $\frac{1}{2403305737672162217073389} a^{16} + \frac{6187619797482238}{64954209126274654515497} a^{13} + \frac{77586685580209149252}{64954209126274654515497} a^{10} - \frac{175791120961966931889}{64954209126274654515497} a^{7} + \frac{830173194499104166559}{1755519165574990662581} a^{4} + \frac{600554625117582313092}{1755519165574990662581} a$, $\frac{1}{1300188404080639759436703449} a^{17} - \frac{10088166759964027505}{1300188404080639759436703449} a^{14} - \frac{9303973555794072664475}{35140227137314588092883877} a^{11} + \frac{1555942742874164116593885}{35140227137314588092883877} a^{8} + \frac{5098375807952316388280289}{35140227137314588092883877} a^{5} - \frac{326806577575338279090058}{949735868576069948456321} a^{2}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{333}\times C_{925407}$, which has order $308160531$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{8100}{1739870348449331} a^{15} - \frac{11850210}{1739870348449331} a^{12} + \frac{4718876041}{1739870348449331} a^{9} - \frac{6573553398}{47023522931063} a^{6} - \frac{537432350346}{47023522931063} a^{3} + \frac{52532403555327}{47023522931063} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 798293358.2555804 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.110889.1, 6.0.36889110963.2, 9.9.110225327118882669776889.10

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ R $18$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ R $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$37$37.9.8.7$x^{9} - 2368$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.7$x^{9} - 2368$$9$$1$$8$$C_9$$[\ ]_{9}$